To draw a frequency polygon for the given data, we follow a series of steps. A frequency polygon is a graphical representation of data points connected by straight lines. Here's a detailed approach to accomplish this:
### Step 1: Identify the class midpoints
The first thing we need to do is determine the midpoints of each class interval. The class midpoint is calculated by taking the average of the upper and lower boundaries of each class interval.
#### Class midpoints calculation:
1. For the interval [tex]\(0 < t \leq 10\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{0 + 10}{2} = 5
\][/tex]
2. For the interval [tex]\(10 < t \leq 20\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{10 + 20}{2} = 15
\][/tex]
3. For the interval [tex]\(20 < t \leq 30\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{20 + 30}{2} = 25
\][/tex]
4. For the interval [tex]\(30 < t \leq 40\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{30 + 40}{2} = 35
\][/tex]
### Step 2: List the frequencies with corresponding midpoints
Next, we create a list of the class midpoints and their corresponding frequencies.
- Midpoints: 5, 15, 25, 35
- Frequencies: 5, 5, 7, 1
### Step 3: Plot the points on the graph
Using the midpoints and frequencies, plot the points on a Cartesian plane where the x-axis represents the midpoints (time) and the y-axis represents the frequency.
### Step 4: Connect the points with straight lines
After plotting the points [tex]\((5, 5)\)[/tex], [tex]\((15, 5)\)[/tex], [tex]\((25, 7)\)[/tex], and [tex]\((35, 1)\)[/tex], connect each point with a straight line to form the frequency polygon.
### Final Graph
Here’s a textual description of the frequency polygon graph:
1. Plot the point (5, 5).
2. Plot the point (15, 5).
3. Plot the point (25, 7).
4. Plot the point (35, 1).
5. Draw straight lines between the points sequentially: from (5, 5) to (15, 5), from (15, 5) to (25, 7), and from (25, 7) to (35, 1).
### Title and Labels
- Title: Frequency Polygon of Times to Complete a Puzzle
- x-axis label: Time (minutes)
- y-axis label: Frequency
By following these steps, you effectively create a frequency polygon that accurately represents the given data.
